Gravity of Two Photon Decay and its Quantum Coherence

A linear analytical solution is derived for the gravitational shock wave produced by a particle of mass $M$ that decays into a pair of null particles. The resulting space-time is shown to be unperturbed and isotropic, except for a discontinuous perturbation on a spherical null shell. Formulae are derived for the perturbation as a function of polar angle, as measured by an observer at the origin observing clocks on a sphere at distance $R$. The effect of the shock is interpreted physically as an instantaneous displacement in time and velocity when the shock passes the clocks. The time displacement is shown to be anisotropic, dominated by a quadrupole harmonic aligned with the particle-decay axis, with a magnitude $\delta \tau\sim GM/c^3$, independent of $R$. The velocity displacement is isotropic. The solution is used to estimate the angular distribution of gravitational perturbations from a quantum state with a superposition of a large number of randomly oriented, statistically isotropic particle decays. This approach is shown to provide a well-controlled approximation to estimate coherent, nonlocal, spacelike correlations of weak-field gravity from systems composed of null point particles up to the Planck energy, including macroscopic quantum coherence of causal quantum-gravitational fluctuations. The solution is extrapolated to estimate gravitational fluctuations from a gas of relativistic particles, and from quantum distortions of a black hole or cosmological horizon.